LINEAR ALGEBRA GUIDE

Matrix methods for solving linear systems

A linear system can be represented as A·x = b. This guide shows when Gaussian elimination, Gauss-Jordan RREF, inverse methods, and Cramer’s rule are useful.

FORMULA

A\mathbf{x} = \mathbf{b}

WORKED EXAMPLE

For 2x + y = 5 and x − y = 1, write A = [[2, 1], [1, −1]], x = [x, y], and b = [5, 1].

STEP BY STEP

Write the coefficient matrix and right-hand side

Keep variables in the same order in every equation. The coefficients form A, the unknowns form x, and the constants form b.

Choose a method that matches the task

Gaussian elimination is efficient for many systems. Gauss-Jordan RREF is useful when you want a reduced row form. The inverse method requires a square, non-singular matrix. Cramer’s rule is most practical for smaller square systems.

Check rank and determinant conditions

A zero determinant means an inverse does not exist. Rank information helps identify whether a system has a unique solution, no solution, or infinitely many solutions.

Verify the final vector

Substitute the output values into the original equations, especially when working with rounded decimals or measured quantities.